Question 1
Which of the following is true ?
Step-by-Step Solution
Detailed Explanation:
√3 $\approx 1.732$ and √7 $\approx 2.645, giving √3 + √7 \approx 4.377$.
$√10 \approx 3.162.$
Since 4.377 > 3.162, the correct inequality is √3 + √7 > √10.
Which of the following is true ?
Detailed Explanation:
√3 $\approx 1.732$ and √7 $\approx 2.645, giving √3 + √7 \approx 4.377$.
$√10 \approx 3.162.$
Since 4.377 > 3.162, the correct inequality is √3 + √7 > √10.
What is the sum to infinity of the series, 3+6 $x^{2}+9$ $x^{4}+12$ $x^{6}$+$\ldots$ given |x| < 1
Detailed Explanation:
Let $S = 3(1 + 2x^2 + 3x^4 + 4x^6 + ...)$.
Using the binomial expansion (1 - y)^{-2} = 1 + 2y + 3y^2 + ... for $y = x^2$:
$S = 3 * (1 - x^2)^{-2} = 3 / (1 - x^2)^2.$
$\lim_{x\rightarrow 0}\frac{\sqrt{1+x}-\sqrt{1-x}}{x}$ is given by
Detailed Explanation:
Rationalize the numerator: [(1+x) - (1-x)] / [x (√(1+x) + √(1-x))] = 2x / [x (√(1+x) + √(1-x))].
Cancel x: 2 / (√(1+x) + √(1-x)).
Taking limit x -> 0 gives 2 / (1 + 1) = 1.
"If \((G, \cdot)\) is a group such that \((ab)^{-1} = a^{-1}b^{-1}, \forall a, b \in G\), then \(G\) is a/an
Detailed Explanation:
In any group, $(ab)^{-1}$ = $b^{-1}$ $a^{-1}$.
Given $(ab)^{-1}$ = $a^{-1}$ $b^{-1}$, we have $b^{-1}$ $a^{-1}$ = $a^{-1}$ $b^{-1}$.
Taking inverse on both sides yields ab = ba, so G is an Abelian group.
A given connected graph G is a Euler Graph if and only if all vertices of G are of
Detailed Explanation:
Euler's Theorem states that a connected graph has an Euler circuit (Euler Graph) if and only if every vertex has an even degree.